Let F be a computable field: a countable field in which the addition and multiplication are given by computable functions. We investigate the Turing degrees d such that F is d-computably categorical, meaning that d is able to compute isomorphisms between F and every other computable field isomorphic to F. We prove that algebraic fields can fail to be 0'-computably categorical, but that there is a degree d, low relative to 0', such that every algebraic field is d-computably categorical. We also prove analogous results, one jump lower, for computable fields F for which the irreducibility of polynomials in F[X] is decidable.